The present invention relates to a method for the interpolative conversion of analog signals to digital signals, wherein the analog input signal is sampled at a multiple of the Nyquist frequency and is quantized with an amplitude resolution corresponding to one bit. The quantized signal is returned to the input and subtracted from the analog input signal. The sampled signal and the quantized signal are converted to a PCM signal by passing through a digital lowpass filter and subsequent sampling at the Nyquist frequency.
A method of the above type is disclosed in J. C. Candy, "A Use of Limit Cycle Oscillations to Obtain Robust Analog-to-Digital Converters", lEEE Trans. Commun., Volume COM-22 (1974), pages 298-305. and in J. C. Candy, Y. C. Ching, D. S. Alexander, "Using Triangular Weighted Interpolation to Get 13-Bit PCM From a Sigma Delta Modulator", IEEE Trans. Commun., Volume COM-24 (1976), pages 1268-1275.
Analog to digital (A/D) converters having an amplitude resolution corresponding to 16 bits per sample value and a sampling rate of 48 kHz are used to digitalize high value audio signals. Conventional A/D converters attain such amplitude resolution by using high precision components. They are not suitable for the integration of a coder PCM for audio signals. Interpolative A/D converters achieve high amplitude resolution not by great precision of the components employed but by sampling at a multiple of the Nyquist frequency and subsequent digital interpolation of the roughly quantized sample values.
FIG. 1 shows the structure of a prior art interpolative A/D converter for audio signals. The analog input signal is initially limited in bandwidth in a lowpass filter A. The difference between the output signal x (t) of the lowpass filter and the feedback signal r (t) is integrated in an integrating network B, sampled at a clock frequency f.sub.s which is a multiple of the Nyquist frequency f.sub.A in sampler C and roughly quantized in an A/D converter D having low amplitude resolution. The output signal of the A/D converter is returned, via a D/A converter E and a holding circuit F, to the input of the integrating network. For the sake of simplicity, the power density spectrum S'.sub.q of the quantizing error q' is initially assumed to correspond to white noise. As shown in FIG. 2, returning the quantized signal through the integrating network B reduces the error component in the low frequency range while it is increased at high frequencies. The spectral composition of the input signal is not changed by this feedback. In a digital lowpass filter G having a bandwidth fg, the high frequency component of the quantizing error q' can now be removed from the signal, so that by subsequent subsampling via a sampler H, a signal can be generated at a lower sampling rate but with reduced quantizing error. Since the input signal is sampled at the Nyquist frequency only at the output of the interpolative A/D converter, the analog channel filter at the input can be omitted. The necessary bandwidth limitation to half the sampling frequency is effected by digital filter G. Since the digital filter and the analog portion of the converter do not require matching and do not contain precision components, the method of interpolative A/D conversion appears to be suitable for the production of a PCM coder for audio signals in the form of an integrated circuit.
A particularly favorable realization of the structure shown in FIG. 1 is obtained if the output signal u' (n.tau.) of the A/D converter D is quantized only with an amplitude resolution corresponding to one bit. The analog component of such a structure can be optimized and the conversion error can be calculated. The discussion below is based on such results.
To facilitate the description of the structure of FIG. 1, reference is made to FIG. 3 where a change is made in the block circuit diagram of FIG. 1 by replacing the 1-bit A/D converter D with a linear amplifier K and an adder for an error signal q (t). For the output signal u' (n .tau.) of the 1-bit A/D converter D, the following then applies in the spectral range: ##EQU1## Where Q (j.omega.) represents system specific error;
X (j.omega.) represents the analog input signal; PA1 A (j.omega.) represents the transfer function of the integrating network; PA1 H (j.omega.) represents the transfer function of the hold circuit
By using A(j.omega.).G&gt;&gt;1 and H (j.omega.)=1 for .omega.&lt;2.pi.W, the following applies: ##EQU2##
Equation (2) shows that the input signal x(t) is not changed in its spectral composition while the power density S'.sub.qu of the error signal contained in u'(n .tau.) results according to the function ##EQU3## With an ideal lowpass filter G having the limit frequency f.sub.g (for a lowpass filter the bandwidth and the limit frequency are equivalent), the remaining residual error P.sub.q of the interpolative A/D converter is calculated, under the above assumptions, at ##EQU4##
For the representative amplitude levels +U and -U of the one-bit A/D converter D, the interpolative A/D converter can be fully modulated with a sinusoidal oscillation of the maximum amplitude U. At the degree of modulation M (O.ltoreq.M.ltoreq.1) of the interpolative A/D converter, the power P.sub.x of a sinusoidal input signal will be ##EQU5##
The signal to noise ratio of the interpolative A/D converter is then calculated as follows: ##EQU6##
By optimizing the transfer function of the integrating network B, the remaining conversion error can be minimized. The previously made assumption that the quantizing error of the one-bit quantizer has a white power density spectrum, can no longer be maintained if the conversion error is analyzed correctly. Rather, it is found that the spectral power density of the quantizing error is dependent on the degree of modulation of the interpolative converter and on the dimensions of the integrating network B and that with decreasing modulation, the conversion error is reduced. The conversion error can be calculated by means of a correction factor k(M) from the value that would result if a white power density spectrum were assumed. With the use of a simple integrator which has the characteristic ##EQU7## and under consideration of a nonwhite power density spectrum S'.sub.q, there results a signal to noise ratio of ##EQU8## where ##EQU9##
Here, k.sub.1 (M) is the correction factor which is dependent upon the modulation M of the interpolative A/D converter and which is shown in FIG. 4. The signal to noise ratio can be improved if instead of a simple integrator a double integrating network is employed which has the characteristic ##EQU10## where .tau.1, .tau.2 and .tau.3 are the time constants of the network, that is shown in FIG. 10. The signal to noise ratio is then calculated as follows: ##EQU11## where ##EQU12##
In order to attain the desired amplitude resolution corresponding to 16 bits in the interpolative A/D converter, Equation (10) indicates that an internal clock pulse frequency F.sub.s of 12 MHz, corresponding to a factor N=256, is necessary for the interpolation loop. Measurement tests made with a laboratory model did not bring the expected results. At a sampling frequency of 48 kHz, the signal to noise ratio realized was only 70 dB. This conversion error which was higher than the calculated value can be traced back only to the non-ideal characteristic of the components employed. The influence of the characteristics of real components on the conversion will be examined below.
In the previous considerations regarding the interpolative A/D conversion, it was assumed, according to FIG. 1, that the train of sampled signals u'(n.tau.) is converted by D/A converter E and holding circuit F in the feedback branch into an exactly meander-shaped signal r(t). Such a signal cannot be realized by way of circuitry. The deviations of the actual signal curve r(t) from the ideal meander shape, as indicated in FIG. 5a, can be traced back to linear and nonlinear distortions in holding circuit F.
The linear distortions caused by holding circuit F can be interpreted essentially as deviations from the optimized transmission behavior of the integrating network B. They can be compensated by appropriate correction of integrating network B.
The nonlinear distortions may be due to the following:
(a) asymmetries in the switching behavior of the holding circuit with the result that the integral over two successive pulses at the same polarity is different from the integral of two individual pulses;
(b) time changes in the length of the loop clock pulse period .tau. (clock jitter) produce an amplitude error in the converted PCM signal; or
(c) inherent noise in the analog components employed reduces the signal to noise ratio of the converter.
Measurements have shown that it is certain that inherent noise is negligible and that only the first two causes require investigation.
The nonlinear distortions caused by asymmetries in the switching behavior of the holding circuit result in a reduction of the intermodulation ratio of the holding circuit. In addition to processing the signal power, the holding circuit must also process high frequency quantizing error components whose power, even under full modulation, is greater than the signal power. An additional error component in the signal frequency range O&lt;f&lt;W is therefore produced as the intermodulation product of the high frequency quantizing error contained in u'(n .tau.). To examine the influence of these asymmetries and check out optimization attempts, the method of interpolative A/D conversion was simulated in a digital computer. In this simulation, a sinusoidal input signal was assumed to exist and the signal to noise ratio of the converter was monitored in dependence on the switching behavior. The switching behavior was initially determined by different rise and fall times for the signal r(t) of FIG. 1. FIG. 6 shows the influence of the difference .delta. between rise time and fall time of the signal r(t) on the signal to noise ratio of the converter. As shown in FIG. 6, in order to attain the required signal to noise ratio of 98 dB, the difference between rise time and fall time of r(t) must not exceed a value of 50 ps.
Time changes in the loop clock pulse period .tau., that may occur due to thermal noise in the active components for clock pulse processing or in the holding circuit, result in an additional error in the interpolative analog to digital conversion. To be able to quantitatively determine this additonal conversion error, the signal r(t) can be resolved, according to FIG. 7, into an exact meander-shaped signal r'(t) and a pulse-shaped error signal q.sub.j (t), where EQU r(t)=r'(t)+q.sub.j (t) (11)
Equation (2) then changes to ##EQU13## where Q(j .omega.) represents the system specific error and Q.sub.j (j .omega.) the realization specific error.
Let it be assumed that .lambda. is a random variable which may take on the value +1 or -1. If it is assumed that the clock pulse period has a Gauss distribution with an average value .tau. and a standard deviation .sigma., then q.sub.j (t) can be approximated by a series of Dirac pulses of the integral 2 U . .sigma. as follows: ##EQU14##
According to the Fourier transformation of (13), the power density spectrum S.sub.qj (f) of the error signal is ##EQU15## and the error power P.sub.qj in the base band is: ##EQU16##
In order for the additional conversion error caused by the time changes in the clock pulse period .tau. to become negligible, the following condition must be met: ##EQU17## For the standard deviation of the clock pulse period .tau. there results a maximum value ##EQU18## which, for the desired signal to noise ratio of 98 dB, must not exceed a value of 6.2 ps.
This theoretically determined value was confirmed by the above described simulation of the interpolative A/D conversion in a digital computer. The switching behavior was here determined by representing the loop clock pulse period .tau. as a random variable having a probability density according to Gauss.
Representation of signal r(t) by meander-shaped pulses having such slight jitter cannot be realized with justifiable expenditures. The method of interpolative A/D conversion was therefore modified with a view toward reducing the errors caused by the realization.
It was thus determined that the accuracy requirements as they exist in today's audio studio techniques with respect to signal to noise ratio cannot be realized with the circuit structure according to the prior art.